Ergodic Behaviour of Nonconventional Ergodic Averages for Commuting Transformations (Journal of Nonlinear Modeling and Analysis, 2024)
Based on T. Tao's celebrated result on the norm convergence ofmultiple ergodic averages for commuting transformations, we find that thereis a subsequence which converges almost everywhere. Meanwhile, we obtainthe ergodic behaviour of diagonal measures, which indicates the time averageequals the space average. According to the classification of transformations,we also give several different results. Additionally, on the torus $\mathbb{T}^d$ with specialrotation, we prove the pointwise convergence in $\mathbb{T}^d$ , and get a result for ergodicbehaviour.
Keywords: Commuting transformation; convergence almost everywhere; ergodic behaviour; time average; space average.